A global regularity result for the 2D Boussinesq equations with critical dissipation

This paper examines the global regularity problem on the two-dimensional incompressible Boussinesq equations with fractional dissipation, given by Λαu in the velocity equation and by Λβθ in the temperature equation, where Λ−−Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda - \sqrt { - \Delta } $$\end{document} denotes the Zygmund operator. We establish the global existence and smoothness of classical solutions when (α, β) is in the critical range: α>(1777−23)/24=0.789103...\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha > (\sqrt {1777} - 23)/24 = 0.789103...$$\end{document}, β > 0, and α + β = 1. This result improves previous work which obtained the global regularity for α>(23−145)/12≈0.9132,β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha > (23-\sqrt {145})/12 \approx 0.9132,\;\beta>0$$\end{document}, and α + β = 1.